Non-equilibrium current and electron pumping in nanostructures

We discuss a numerical method to study electron transport in mesoscopic devices out of equilibrium. The method is based on the solution of operator equations of motion, using efficient Chebyshev time propagation techniques. Its peculiar feature is the propagation of operators backwards in time. In this way the resource consumption scales linearly with the number of states used to represent the system. This allows us to calculate the current for non-interacting electrons in large one-, twoand three-dimensional lead-device configurations with time-dependent voltages or potentials. We discuss the technical aspects of the method and present results for an electron pump device and a disordered system, where we find transient behaviour that exists for a very long time and may be accessible to experiments. Electron transport through mesoscopic devices contacted to leads is intensely studied in chemistry and physics (see e.g. Refs. [1, 2]). The conceptual basis for theoretical studies is the non-equilibrium Green function (Keldysh) formalism [3]. The Meir-Wingreen-formula [4] allows for the calculation of steady state currents. For time-dependent potentials or gate voltages the calculation of a current is a serious problem already for non-interacting electrons. One crucial point in most approaches is that the resource consumption scales quadratically with the number of system sites, since each Green function Gij(t, t ′) or operator product c†i (t)cj(t ′) has two site indices. The necessity to study large systems conflicts with the rapid growth of computational demands, especially if long leads with long recurrence time are required. Recent studies therefore addressed mainly one-dimensional (1D) situations, e.g. by time-propagation of operator expectation values or of several single-particle eigenstates [5–7]. In the present contribution we numerically solve the equation of motion for a single fermion operator. In contrast to previous studies, the initial time coordinate is propagated backwards in time. Then the computational effort, in particular the memory consumption, scales linearly with the number of lattice sites used to represent device and leads. This allows us to calculate the non-equilibrium current even for large systems that could not be treated otherwise. We consider the following situation: A device of Lx × Ly × Lz-sites is contacted to two long leads extending along the x-direction (see figure 1). This slab geometry includes the 2D (Lz = 1) and 1D (Ly = Lz = 1) case. For the kinetic energy in the Hamiltonian